面滾式傘齒輪加工技術由於精度高、生產效率高，是汽車行業中量產製造戟齒輪的主要方法。它和面銑式加工技術皆可於新式六軸CNC傘齒輪切齒機上進行製造。然而，這台機器具備幾項關鍵技術：例如非線性六軸NC路徑編程、切削模擬和切削力預測等技術；國內目前沒有製造商具備這些關鍵技術，因此他們無法生產這類機器。另外，業界最常用的市售切削模擬軟體VERICUT僅能分析一般端銑刀的切削力，無法用於分析量產齒輪專用刀具。換句話說，VERICUT無法用於計算大部分齒輪加工的切削力。
本研究主要目的是在建立基於三維環線法之傘齒輪面滾式切削模擬數學模式，透過逆向運動學，推導新式六軸CNC傘齒輪切齒機的機械設定，並可以進一步建立其切屑的數學模式。接著我們就能根據切屑體積和進給速率去計算體積移除率(MRR)，因此這些研究可以作為開發CNC傘齒輪切齒機的關鍵技術。
本研究是第一個在新式六軸CNC傘齒輪切齒機上完成面滾式加工實驗。透過分析體積移除率和刀軸扭矩之間的關聯性，我們可以優化切削進給速率，在小齒輪創成法粗加工節省約25%的加工時間，總計花費2分8秒完成加工；根據小齒輪的加工經驗，進行大齒輪的優化實驗，最終花費1分10秒加工大端模數4.88mm的大齒輪，並得到DIN4級的切削結果，完美展現出面滾式加工的高效率和高精度。

Face hobbing is the auto industry's main mass production method for bevel gears due to its high precision and productivity. It and face-milling method can be implemented on the new six-axis CNC bevel gear machine. However, this machine has several key technologies, for example, non-linear six-axis NC codes programming, cutting simulation, prediction of cutting force, and so on. Up to now, domestic machine manufacturers do not have these key technologies. Therefore, they could not produce this machine. Besides, the most popular cutting simulation software VERICUT can only simulate end-mill cutting forces but not special purpose tools. In other words, this commercial software cannot calculate the cutting forces of most gear machining.
This research proposes a simulation method for face hobbing cutting based on the three-dimensional circle method. Six-axis cutting coordinates of face hobbing are derived based on inverse kinematics. And the mathematical model of its cutting chips can be further established. Moreover, we can calculate the volume removal rate (MRR) from chip volume and feed rate. Therefore, these researches can be used to develop the key technologies of CNC bevel gear cutting machines.
This research is the first to finish face-hobbing experiments on a new six-axis CNC bevel gear machine. We optimize the cutting feed rate to reduce cutting time of pinion up to 25% by analyzing the correlation between MRR and tool torque, which take a total of 2 minutes 8 seconds to complete the machining. According to the processing experience of pinion, we took 1 minute and 10 seconds to process gears with heel module of 4.88 mm, and the cutting results of DIN4 were obtained, which perfectly demonstrated the high efficiency and precision of face hobbing.

[1] Artzy, E., Frieder, G. and Herman, G. T., 1981, "The theory, design, implementation and evaluation of a three-dimensional surface detection algorithm," Computer Graphics and Image Processing 1981 Vol.15 pp. 1-24
[2] Lorensen, W. E., and Cline, H. E., 1987, "Marching cubes: A high resolution 3D surface construction algorithm," Proc. Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques, pp. 163-169.
[3] Gottschalk, S., Lin, M. C., and Manocha, D., 1996, "OBBTree: A hierarchical structure for rapid interference detection," Proceedings of the 1996 Computer Graphics Conference, SIGGRAPH, Anon, ed. New Orleans, LA, USA, pp. 171-180.
[4] Eberly, D., 2001, "Dynamic collision detection using oriented bounding boxes," Mathematics.
[5] Van Hook, T., 1986, "Real-time shaded NC milling display," 13th Annual Conference on Computer Graphics and Interactive Techniques., pp. 15-20.
[6] Zhang, W., Peng, X., Leu, M. C., and Zhang, W., 2007, "A novel contour generation algorithm for surface reconstruction from dexel data," Journal. Computer. Inf. Sci. Eng., 7(3), pp. 203-210.
[7] Böß, V., Denkena, B., Breidenstein, B., Dittrich, M. A., and Nguyen, H. N., 2019, "Improving technological machining simulation by tailored workpiece models and kinematics," 17th CIRP Conference on Modelling of Machining Operations, CIRP CMMO, E. Ozturk, T. McLeay, and R. Msaoubi, eds., Elsevier B.V., pp. 224-230.
[8] Inui, M., Huang, Y., Onozuka, H., and Umezu, N., 2020, "Geometric simulation of power skiving of internal gear using solid model with triple-dexel representation," 48th SME North American Manufacturing Research Conference, NAMRC 48, L. Fratini, I. Ragai, and L. Wang, eds., Elsevier B.V., pp. 520-527.
[9] Katz, A., 2017, "Cutting Mechanics of the Gear Shaping Process," UWSpace.
[10] 丁銓峰，2021，基於三維環線法之傘齒輪面銑式切削模擬，碩士，國立台灣科技大學，台北市。
[11] Warkentin, A., Ismail, F., and Bedi, S., 1998, "Intersection approach to multi-point machining of sculptured surfaces," Computer. Aided Geom. Des., 15(6), pp. 567-584.
[12] Warkentin, A., Ismail, F., and Bedi, S., 2000, "Multi-point tool positioning strategy for 5-axis machining of sculptured surfaces," Computer. Aided Geom. Des., 17(1), pp. 83-100.
[13] Yanwei, X., Lianhong, Z., Wei, W., and Leping, W., 2009, "Virtual simulation machining on spiral bevel gear with new type spiral bevel gear milling machine," 2009 International Conference on Measuring Technology and Mechatronics Automation, ICMTMA 2009Zhangjiajie, Hunan, pp. 432-435.
[14] Wang, Y.,Wang, T. Y.,Lin, F. X.,Lu, Z. L.,Wang, D., 2013, "The research of the epicycloids bevel gear cutting based on the common six-axis machine," Advanced Materials Research, Vol.819, pp. 110-114.
[15] Altintas, Y., and Merdol, S. D., 2007, "Virtual high performance milling," CIRP Ann Manuf. Technol., 56(1), pp. 81-84.
[16] Merdol, S. D., and Altintas, Y., 2008, "Virtual simulation and optimization of milling applications - Part II: Optimization and feedrate scheduling," J Manuf. Sci. Eng. Trans. ASME, 130(5), pp. 0510051-05100510.S
[17] Chen, S. H., and Fong, Z. H., 2015, "Study on the cutting time of the hypoid gear tooth flank," Mechanism and Machine Theory, 84, pp. 113-124.
[18] Fu, H. J., DeVor, R. E., and Kapoor, S. G., 1984, " Mechanistic model for the prediction of the force system in face milling operations," Journal of Engineering for Industry ASME, 106(1), pp. 81-88.
[19] Suzuki, T., Kondo, S., and Ueno, T., 1985, " Study on the cutting forces for spiral bevel gears," Bull JSME, 28(240), pp. 1316-1323.
[20] Kundrák, J., Markopoulos, A. P., Makkai, T., Deszpoth, I., and Nagy, A., 2018, "Analysis of the effect of feed on chip size ratio and cutting forces in face milling for various cutting speeds," Manuf. Technol., 18(3), pp. 431-438.
[21] Onozuka, H., Tayama, F., Huang, Y., and Inui, M., 2020, "Cutting force model for power skiving of internal gear," J. Manuf. Processes, 56, pp. 1277-1285.
[22] Zheng, F., Han, X., Hua, L., Tan, R., and Zhang, W., 2021, "A semi-analytical model for cutting force prediction in face-milling of spiral bevel gears," Mechanism and Machine Theory, 156.
[23] Zheng, F., Han, X., Lin, H., and Zhao, W., 2021, "Research on the cutting dynamics for face-milling of spiral bevel gears," Mechanical. Syst. Signal Process, 153.
[24] ANSI/AGMA ISO 23509-A08, 2008, “Bevel and Hypoid Gear Geometry”, Alexandria, USA.
[25] Litvin, F., and Gutman, Y., 1981, "Methods of Synthesis and Analysis for Hypoid Gear-Drives of “Formate” and “Helixform”—Part 1. Calculations For Machine Settings For Member Gear Manufacture of the Formate and Helixform Hypoid Gears " Journal of Mechanical Design, 103, pp. 89-101.
[26] Litvin, F. L., Zhang, Y., Lundy, M., and Heine, C., 1988, "Determination of settings of a tilted head cutter for generation of hypoid and spiral bevel gears," Journal of Mechanisms Transmissions and Automation in Design, 110(4), pp. 495-500.
[27] 董學朱，2002，擺線齒錐齒輪及準雙曲面齒輪設計和製造，機械工業出版社，北京。
[28] Fong, Z. H., and Tsay, C. B., 1991, "A Mathematical Model for the Tooth Geometry of Circular-Cut Spiral Bevel Gears," Journal of Mechanical Design, 113(2), pp. 174-181.
[29] Fong, Z. H., 2000, "Mathematical Model of Universal Hypoid Generator With Supplemental Kinematic Flank Correction Motions," Journal of Mechanical Design, 122(1), pp. 136-142.
[30] Shih, Y. P., Fong, Z. H., and Lin, G. C. Y., 2006, "Mathematical Model for a Universal Face Hobbing Hypoid Gear Generator," Journal of Mechanical Design, 129(1), pp. 38-47.
[31] Shih, Y. P., and Fong, Z. H., 2008, "Flank correction for spiral bevel and hypoid gears on a six-Axis CNC hypoid generator," Journal of Mechanical Design, Transaction ASME, 130(6), pp. 0626041-06260411.
[32] Fan, Q., Dafoe, R. S., and Swanger, J. W., 2008, "Higher-order tooth flank form error correction for face-milled spiral bevel and hypoid gears," Journal of Mechanical Design, Transaction ASME, 130(7), pp. 0726011-0726017.
[33] Fan, Q., 2010, "Tooth surface error correction for face-hobbed hypoid gears," Journal of Mechanical Design, Transaction ASME, 132(1), pp. 0110041-0110048.
[34] Shih, Y. P., Sun, Z. H., and Lai, K. L., 2017, "A flank correction face-milling method for bevel gears using a five-axis CNC machine," Int. J Adv. Manuf. Technol., 91(9-12), pp. 3635-3652.
[35] Hsu, R. H., Shih, Y. P., Fong, Z. H., Huang, C. L., Chen, S. H., Chen, S. S., Lee, Y. H., and Chen, K. H., 2020, "Mathematical model of a vertical six-axis Cartesian computer numerical control machine for producing face-milled and face-hobbed bevel gears," Journal of Mechanical Design, Transaction ASME, 142(4).
[36] 蘇宏旻，2013，面滾式直傘齒輪SolidWorks API切削模擬，碩士，國立台灣科技大學，台北市。
[37] Gleason, 2000, Face Hobbing Development, Rev. 1.0, pp. 2-23